Issue 98034 | Editor: Erik Sandewall | 13.4.1998 |
Today |
The discussion about ontology of time continues with an additional contribution by Pat Hayes. This discussion has so far been included under the general discussion on the topic of "ontologies for actions and change", but now it seems more appropriate to spin it off into a topic of its own. This is motivated both by the number of significant contributions that have been made, and by the general importance of this topic.
Thus, for readers of the E-mailed newsletter, everything proceeds as before, but the structured discussion record in our webpage structure has been revised retroactively to correspond to the new classification.
The present issue also contains a question by Rob Miller to Erik Sandewall re his ETAI submitted article.
ETAI Publications |
Additional contributions have been received for the discussions about the following article(s). Please click the title of the article in order to see each contribution in its context.
Erik Sandewall
Logic-Based Modelling of Goal-Directed Behavior
Debates |
Answer to Jixin's contribution to this discussion on 1.4:
There is really little point in arguing about which theories are more 'intuitive' unless one is more precise about what one's intuitions are. There are two fundamental problems with arguments like this. First, intuitions are malleable, and one can get used to various ways of thinking about time (and no doubt many other topics) so that they seem 'intuitive'. Second, our untutored intuition seems to be quite able to work with different pictures of time which are in fact incompatible with one another. Jixin's own intuition, for example, seems to agree with McCarthy's that time is continuous, and yet also finds the idea of contiguous atomic 'moments' (intervals with no interior points) quite acceptable. But you can't have it both ways: if moments can meet each other, then there might not be a point where the speed is exactly 60mph, or the ball is exactly at the top of the trajectory. If time itself is discrete, then the idea of continuous change is meaningless. Appealing to a kind of raw intuition to decide what axioms 'feel' right lands one in contradictions. (That was one motivation for the axiom in Allen's and my theory, which Jixin found "awkward", that moments could not meet. The other was wanting to be able to treat moments as being pointlike. That was a mistake, I'll happily concede.)
Jixin wrote:
What follows is our response to the arguments about the ontology of
time from Pat Hayes, Ray Reiter, and John McCarthy.
Response To John: The example of car accelerating demonstrates the need of time points for time ontology. A similar example is throwing a ball up into the air. The motion of the ball can be modelled by a quantity space of three elements: going-up, stationary, and going-down. Intuitively, there are intervals for going up and going down. However, there is no interval, no matter how small, over which the ball is neither going up nor going down. The property of being stationary is naturally associated with a point, rather than any interval (including Allen and Hayes' moment), a "landmark" point which separates two other intervals.
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Response To Ray: (---)
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Response To Pat: |
...For example, the 1990 AIJ critique of Allen's account by Galton (wrongly) assumes that Allen's intervals are sets of points on the real line. |
After re-reading Galton's paper
[j-aij-42-159],
as we understand, Galton's
arguments are in general based on the assumption that Allen's
intervals are primitive, rather than sets of points on the real line.
In fact, the main revision Galton proposes to Allen's theory is a
diversification of the temporal ontology to include both intervals and
points. That is, in Galton's revised theory, intervals are still
taken as primitive.
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Having pointed out this, however, as shown in Ma, Knight and Petrides' 1994 paper [j-cj-37-114], Galton's determination to define points in terms of the "meeting places" of intervals does not, as he claims, axiomatise points on the same footing as intervals, and hence that some problems still remain in these revisions. |
there does seem to be a simple, basic, account which can be extended in various ways to produce all the other alternatives, and this core theory is the one I was referring to. |
Does this core theory refer that one in which "intervals are uniquely
defined by their endpoints (which are also the points they fit
between) and two intervals meet just when the endpoint of the first is
the startpoint of the second"? Or Allen's one? - It seems the former
one.
Anyway, yes. There does seem to be such a simple, basic core theory.
For general treatments, in Ma and Knight's CJ94 paper
[j-cj-37-114], a time theory
is proposed (as an extention to Allen and Hayes' interval-based
one) which takes both intervals and points as primitive on the same
footing - neither intervals have to be constructed out of points, nor
points have to be created as the places where intervals meet each
other, or as some limiting construction of intervals. The temporal
order is simply characterised in terms of a single relation "Meets"
between intervals/points.
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One technical point, about 'primitive'. One of the things I realised when working with James on this stuff was that if ones axioms about points were minimally adequate it was trivial to define interval in terms of points; and one can also define points in terms of intervals, although that construction is less obvious. (I was immensely pleased with it until being told that it was well-known in algebra, and first described by A. N. Whitehead around 1910.) Moreover, these definitions are mutually transparent, in the sense that if one starts with points, defines intervals, then redefines points, one gets an isomorphic model; and vice versa. So to argue about which of points or intervals are 'primitive' seems rather pointless. We need them both in our ontology. If one likes conceptual sparseness, one can make either one rest on the other as a foundation; or one can declare that they are both 'primitive'. It makes no real difference to anything.
Some advantages of this time theory are:
(1) It retains Allen's appealing characteristics of treating
intervals as primitive which overcomes the Dividing Instant Problem.
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(2) It includes time points into the temporal ontology and therefore
makes it possible to express some instantaneous phenomenon, and
adequate and convenient for reasoning correctly about continuous
change. (3) It is so basic that it can be specified in various ways to
subsume others. For instance, one may simply take the set of points
as empty to get Allen's interval time theory, or specify each
interval, say T, as
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....
Yes, it's true. And, it seems that, all these can be reached
equivalently by simply taking pointlike interval
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one can, for example, say something like
((illuminated i) or (dark i))) implies (not (pointlike i))so that the light is neither on nor off at the switching point. In this theory, every proposition has a 'reference interval' during which it is true, and a proposition might not be true of subintervals of its reference interval. (Though some propositions might be. This kind of distinction has often been made in the linguistic literature. Note however that this intuition is basically incompatible with the idea that an interval is identical to the set of the points it contains.) |
This can be distinguished by applying
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....Actually, as shown in Ma and
Knight's 1996 paper
[j-cj-37-114], to characterise the intuitive relationship
between
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Also, it seems that, in Pat's formulation, for expressing that
interval
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You can consistently add that rising and falling are true for all nonpointlike subintervals and every properly contained subinterval of the reference interval.
On the one hand, many cases suggest the need of allowing a proposition
to holds at a single point. For instance, see the example of throwing
a ball up into the air described earlier in the response to John
MaCarthy.
On the other hand, allowing a proposition to holds at a single point doesn't necessarily make pointlike un-definable. It depends on if one would impose some extra constraints, such as ((illuminated i) or (dark i))) implies (not (pointlike i))as introduced by Pat for the light switching example, which actually leads to the assertion that the light is neither on nor off AT the switching point. |
In my theory it leads to the conclusion that
Actually, in the later version of Allen and Hayes's theory that appears in
1989
[j-ci-5-225], an awkward axiom is proposed to
forbid moments to meet each other. It is interesting to note that,
although moments are quite like points (moments are non-decomposable),
they still have positive duration (they are not pointlike). Moments
are included in Allen and Hayes' time ontology, while points are not.
One of the reason that such an axiom is awkward is that it doesn't
catch the intuition in common-sense using of time.
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But these cases only make sense if one thinks of interval and points in the usual mathematical way, which is exactly what Im suggesting we don't need to do. We can get almost everything we need just from the ordering structure: we don't need to get all tied up in distinguishing cases which can only be formally stated by using all the machinery of real analysis. |
The cases make sense not only if one thinks of intervals and points in
the usual mathematical way. In fact, all the three cases are
demonstrated under the assumption that both intervals and points are
treated as primitive, rather than in the usual mathematical way.
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... Is the light on or off at (exactly) 3.00 pm? The only way to answer this is to find a suitable non-pointlike interval of light or darkness completely surrounding 3.00 pm, because 'being on' is the kind of proposition that requires a nonpointlike reference interval. |
But it seems that there are also some other kind of proposition to
which one cannot assign any nonpointlike reference interval. For
instance, in the throwing ball up into the air example, proposition
"the ball is stationary" can only be true at points, and for any point
we cannot find any non-pointlike interval (completely) surrounding it
over which the ball is stationary.
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This has nothing to do with whether an interval is open or closed:
in fact, there is no such distinction in this theory. It only
arises in a much more complicated extension which includes set
theory and an extensionality axiom for intervals.
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In the time theory where both intervals and points are taken as
primitive, we can (if we like) talk about the open and closed nature
of intervals with some knowledge being available. This kind of knowledge
can be given in terms of the Meets relation, rather than some "much
more complicated extension which includes set theory and an
extensionality axiom for intervals". In fact, we can define that:
It is important to note that the above definition about the open and closed nature of intervals is given in terms of only the knowledge of the Meets relation. However, if one would like to specify intervals as point-based ones, such a definition will be in agreement with the conventional definition about the open and closed intervals. |
That certainly seems to be an elegant device. (Though the definitions have
nothing to do with knowledge; all Jixin is saying is that the definitions
of open and closed can be given in terms of
On the other hand, if the theory allows distinct points to
Pat Hayes
PS. Maybe the most useful thing would be to put all these axiomatic theories into some common place with a common syntax - we could use vanilla-KIF - so people can compare and contrast them. I dont have enough, er, time to offer to do this right now, im afraid, but will cooperate with anyone who will volunteer.